# Linear independence and orthogonality of vectors

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• Jan 9th 2013, 09:08 AM
guybrush92
Linear independence and orthogonality of vectors
okay, so this is really two questions, the first is kind of simple:

if a+b=0, a+c=0 and c=b can it be shown that a+b+c=0? (without the use of determinants)

The second is: what is an example of a set of vectors that are linearly independent but not orthogonal? (graphical interpretation would be appreciated)
• Jan 9th 2013, 09:41 AM
GJA
Re: Linear independence and orthogonality of vectors
Hi guybrush92,

Knowing \$\displaystyle a + b = 0\$, \$\displaystyle a+c=0\$ and \$\displaystyle b=c\$ (which follows from the first two equations) it does not always follow that \$\displaystyle a+b+c=0.\$ Try playing around with some specific values for \$\displaystyle a, b \& c\$ to see why.

For the second question, thinking about \$\displaystyle \mathbb{R}^{2}\$ might be the easiest place to look. Geometrically, two vectors are linearly indepdent in \$\displaystyle \mathbb{R}^{2}\$ if they don't lie on the same line through the origin. It may be simplest to take one of the vectors to be \$\displaystyle [1,0].\$ To find a second vector that is linearly independent from \$\displaystyle [1,0]\$ and is not orthogonal to it means we should pick a vector that does not sit on the \$\displaystyle x\$-axis and is not perpendicular to the \$\displaystyle x\$-axis.

Does this get things going in the right direction? Let me know if anything is unclear. Good luck!